Introduction to General Equilibrium Juan Manuel Puerta November 6, 2009
Introduction So far we discussed markets in isolation. We studied the quantities and welfare that results under different assumptions on market power. The analysis of markets in isolation is referred as partial equilibrium. In contrast, General Equilibrium analysis, studies the interactions of all the markets. In this case, all the prices are variable and adjust. Equilibrium requires that all the markets clear. General equilibrium could be studied considering or not the production of goods. We will study general equilibrium without production and focusing on the exchange of goods.
Some Definitions There are n consumers in this economy. A consumer is described by her utility function u i and her initial endowment ω i = {ω i,1,..., ω i,k } for each of the k goods of this economy. The Consumption bundle of the consumers is given by x i = {xi 1,..., xk i } and describes how much of each good is consumed. An Allocation is the collection of the n consumption bundles. In other words an allocation is an n k vector x = {x 1,..., x n } An allocation is feasible if the sum of consumption bundles is less than the sum of endowments. That is, n n x i ω i (1) In a 2 2-framework (i.e. 2 goods, 2 consumers), allocations can be easily depicted with an Edgeworth box.
Some Definitions Let m i denote the consumer s income, i.e. the market value of her endowment. m i = pω i The consumer problem is given by: max x i u i (x i ) (2) subject to m i = px i We studied that the solution to this problem is given by a function 1 x i = x i (p, m i ). Note that unlike the standard case in the consumer theory, m i = pω i is a function of prices! 1 Assuming strict convexity of preferences, the solution is unique
Walrasian Equilibrium: Definition Walrasian Equilibrium: The pair (x i, p ) is a Walrasian equilibrium if n n x i (p, m i ) ω i (3) that is, if the amounts demanded are less or equal than the amount supplied for each good. No good is excess demand. Why do we allow the possibility of excess supply? It could be the case that a good is undesirable, so there is excess supply of it.
Excess Demand Function: Properties Under which conditions can we assure that there exist a p that makes market to clear? In order to find an answer to this question it is useful to define the excess demand function z(p) z(p) = n [x i (p, pω i ) ω i ] Properties of z(p) 1 Homogeneous of degree 0. From consumer theory, the demand functions are homogeneous of degree 0 in prices, i.e. x i (p, pω i ) = x i (kp, kpω i ). This property carries on to the excess demand function (z(p)), defined as where we ignore the fact that z(p) depends on ω i, as the initial endowments remain constant. 2 Continuity. If all x i are continuous, so is z(p) 3 Walras Law. For any price vector p, pz(p) = 0, that is, the value of the excess demand is identically 0. Proof: The BC imply px i (p, pω i ) = pω i ). Sum across individuals and rearrange.
Market Clearing. If demands equals supply in k-1 markets and p k > 0, then demand equal supply in the k-th market. Proof: Let z j denote the j-th element of the z(p) vector. Walras law implies j p j z j (p) = 0. Rewrite as j k p j z j (p) + p k z k (p) = 0. If all but the k-th market clear, then p j z j (p) = 0 for j k. Then Walras Law implies that p k z k (p) = 0, i.e. z k (p) = 0 as p k > 0. Free Goods. If p is a Walrasian Equilibrium and z(p ) 0, then p j = 0. That is, if some good is in excess supply at the Walrasian Equilibrium, it must be a free good. If p is a Walrasian Equilibrium, z(p ) 0. Since prices are non-negative, p z(p ) = k p k z k(p ) 0. Then either the market clears z k (p ) = 0 or if z k (p ) < 0, p k = 0. Otherwise k p k z k (p ) < 0 and the walras law does not hold.
Definition: Desirability. We say a good i is desirable if p i = 0 implies z i (p) > 0 Idea: A good is desirable if when it is free, there is excess demand of it! It turns out that desirability allows us to find an important result, equality between demand and supply. Equality between demand and supply. If all goods are desirable and p is a price vector consistent with Walrasian equilibrium, then z(p ) = 0, that is, in equilibrium, supply equals demand in every market. Proof: Assume not. Then z(p ) < 0. If that is the case then there is some i for which p i = 0. Desirability then implies z i (p ) > 0, a contradiction.
Summary In equilibrium we will expect markets to clear or to show excess supply. If a market is in excess supply, then that means that the good has to have a zero price in equilibrium. If we assume that all goods are desirable in the sense that people want to consume more than what is available at price zero, then all markets should clear in equilibrium.
Existence Since z(p) is HD0, we can normalize prices and express everything in terms of relative prices. A convenient normalization of the absolute prices ˆp i is given by p i = ˆp i k j=1 ˆp j (4) This implies that p i sum up to 1. So, let k be the number of goods in this economy, we can restrict our attention to the following set of normalized prices S k 1, the k-1 unit simplex Example: S 1 and S 2 S k 1 = {p R k + : k p k = 1}
Brouwer s Fixed Point Theorem In order to prove existence of general equilibrium we need first a couple of mathematical results. Intermediate Value Theorem: Let f(x) be a continuous function defined over [a,b]. Then, for every d between f(a) and f(b), there exist a c such that d=f(c). Brouwer s Fixed Point Theorem Let f : S k 1 S k 1 be a continuous function from the k-1 unit simplex into itself, there is some x in S k 1 such that f(x) = x. Proof for the one dimensional case. Let f : [0, 1] [0, 1]. Let g(x) = f (x) x. Now, g(0) = f (0) 0 0 and g(1) = f (1) 1 0. By the intermediate value theorem, there exist x such that g(x) = 0. But this implies f(x)=x establishing the result. graphical intuition. For a more general proof in the simplex see Starr (p.56-63) (Beyond the scope of this class).
Proof of existence I Theorem Existence of Walrasian Equilibria. If z : S k 1 R k is a continuous function that satisfies Walras Law, pz(p) 0, then there exist some p S k 1 such that z(p ) 0 Proof: Define a map g : S k 1 S k 1 by g i (p) = p i + max{0, z i (p)} 1 + for,2,...,k (5) k j=1 max{0, z j (p)} Note that g i (p) is continuous as both z(p) and the max function are continuous.note also that g(p) is a point in the simplex since gi (p) = 1.There is also a economic interpretation for this. The relative price of goods in excess demand (z i > 0) is increased, so as to
Proof of existence II eliminate excess demand.now, Brouwer s fixed point theorem ensures that there exists p such that p = g(p ). That is p i = p i + max{0, z i (p )} 1 + for,2,...,k (6) k j=1 max{0, z j (p )} The last part of the proof requires that we show that the vector p consistent (6) is associated with a Walrasian equilibrium. In order to so, just rearrange the expression p i + p i k max{0, z j (p )}) = p i + max{0, z i (p )}, for,2,...,k j=1
Proof of existence III Multiply these equations by z i (p ) and sum up across goods we obtain k k [ max{0, z j (p )})] p i z i(p ) = j=1 k z i (p ) max{0, z i (p )}) By Walras Law k p i z i(p ) = 0, so the expression is simplified to k z i (p ) max{0, z i (p )}) = 0 Which requires that z i (p ) 0 for all i. This means that p is the price vector consistent with a walrasian equilibrium.
Implications of the Theorem The existence of general equilibrium could be proved under very general conditions. We specifically had to assume just that: 1 Continuity of z(p): A sufficient condition for this is that individual demand functions are continuous. This requires us to assume strict convexity of preferences, then the demand correspondence is single-valued (it is a function) and it is continuous by the theorem of the maximum. But continuity of the aggregate demand may still be achieved by aggregation of non-continuous individual demand functions. 2 Walras Law: This property follows from the fact that individuals are faced with some form of budget constraint. There is a technical issue though. This is the assumption of continuity may break down as p i 0. As the price of something that is actually desired goes to zero, excess demand may explode. This point is overcome with a slightly more complicated mathematical proof.
Welfare Theorems While it is important to establish the existence of Walrasian equilibrium under broad circumstances, we have not been able to say anything normative about this equilibrium? Is it good? Could it be improved? In order to answer these questions we have to ask what do we mean by good and improve. Definition (Pareto Efficiency) A feasible allocation x is weakly Pareto efficient allocation if there is no feasible allocation x such that all the agents strictly prefer x to x. It is strongly Pareto efficient if all agents weakly prefer x to x and there is at least one agent who strictly prefers x to x.
Theorem (Pareto Efficiency) If preferences are continuous and monotonic, then an allocation is weakly Pareto efficient if, and only if, the allocation is strongly Pareto Efficient. That is, the two definitions are equivalent. Proof: Clearly SPE implies WPE (if you cannot improve 1 agent without hurting other one, then you simply cannot improve all the agents). The converse statement is not difficult to prove. If you have 1 person that is strictly better off (x i ). You can take a small amount (1 θ)x i and redistribute it to all the other (n-1) agents giving (1 θ)x i /(n 1) to each. Continuity implies that we can choose θ close enough to 1 so that agent i still better off. Monotonicity implies that all the rest are also better off, so that the allocation is strongly pareto efficient.
A pareto efficient allocation solves de maximization problem Edgeworth box max 1(x 1 ) {x 1,x 2 } such that u 2 (x 2 ) ū x 1 + x 2 = ω 1 + ω 2 (7) It should be clear that Pareto efficiency requires tangency of indifference curves. The locus of points of all the Pareto efficient allocations is the contract curve
Note that a walrasian equilibrium equalizes the price ratios to the MRS of a consumer. Since all consumers face the same walrasian prices, then this amounts to equalizing marginal rates of substitution across individuals. Note that the solution to problem (7) also involves equalization of marginal rates of substitution. Before we continue, it would be useful to define precisely what we mean by Walrasian equilibrium. We will assume desirability in order to simplify the argument to come. Definition (Walrasian Equilibrium) An allocation-price pair (x, p) is a Walrasian Equilibrium if the allocation is feasible and each agent is maximizing his utility given his budget set. That is, n 1 Feasible: x i = n ω i 2 Ut. Max: x i solves the UMP of agent i. That is, if x i x i then px i > pω i
Is there a correspondence between Walrasian equilibria and the set of Pareto optima? It turns out that there is. In the next two theorems will establish a relationship. Roughly speaking, 1 The first welfare theorem says that every Walrasian equilibrium is Pareto optimum. 2 The second welfare theorem proves the converse statement. For every Pareto optimum, there exist a set of prices and endownments such that it can be supported as a Walrasian equilibrium.
First Welfare Theorem Theorem First Theorem of Welfare Economics. If (x, p) is a Walrasian equilibrium, then x is Pareto efficient. Proof. Assume not and let x be a feasible allocation that all agents prefer to x. Then the definition of walrasian equilibrium implies that all agents are maximizing utility, so since they prefer x to x, px i > pω i. Sum up across individuals to get, n px i > n pω i. But this contradicts the feasibility assumption. The contradiction establishes the result.
Second Welfare Theorem I Theorem Second Theorem of Welfare Economics. Assume x is a Pareto Efficient allocation in which each agent holds a positive amount of each good. Suppose that preferences are convex, continuous and monotonic. Then x is a Walrasian equilibrium for the initial endowments ω i = x i for i = 1, 2,..., n. Proof: Let P i = {x i R k : x i i x i }. P i is the set of bundles that agent i prefers to x i. Define P = i P i = {z : z = i x i, x i P i }. Since each P i is convex by assumption and the sum of convex sets is convex. Then P is convex. Before we go on we need to state a theorem that is useful.
Second Welfare Theorem II Theorem (Separating Hyperplane) If A and B are two nonempty, disjoint, convex sets in R n, then there exist a p 0 such that px py for all x A and y B. Let ω = n x i.pareto optimality of allocation ω means that ω is not in P. Since P is nonempty and convex and it is disjoint with the set that includes only the element ω, we can use the separating hyperplane theorem pz p n x i for all z P
Second Welfare Theorem III In order to finalize the proof, we need to establish that, 1) p is a price vector, that is, it is non-negative, 2) every agent is maximizing utility, that is, if there is a bundle that they prefer to x i, it should be more expensive than x i. 1. p 0: Let e i = (0,..., 1,..., 0) where the one is in the i-th position. ω + e i P since it has one more unit of the good i, so everyone could be made better off. But then, the separating hyperplane theorem implies p( ω + e i ω) = p i 0. Since you can do this for any i, the non-negativity of prices follows. In order to establish that show that strictly preferred bundles are more expensive we do it in 2 steps. First we show that they must be at least as expensive and use that to prove the strict inequality. 2. if y j j x j then py j > px j : We need to show if a particular agent j prefers y j to x j, then y j is never cheaper. The trick is to redistribute the extra good that j got among all the other consumers using strong
Second Welfare Theorem IV monotonicity and continuity. Then, we can use the separating hyperplane theorem and we are done. Define, z j = (1 θ)y j z i = x i + θ n 1 y j, for i j Now this new allocation i z i is in P as everyone is strictly better off than before. Using the separating hyperplane theorem, p z i p But replacing the definitions of z i above, i j i p[(1 θ)y j z i x θ i + n 1 y j] p[xj + x i ] i j i x i i j
Second Welfare Theorem V p[(1 θ)y j + θy j + x i ] p[x j + x i ] i j i j py j px j Now the last step is to prove that this result actually holds with strict inequality. Continuity of preferences implies that we can find θ sufficiently close to 1 such that θy j j x j. By the argument just presented, it is possible to show that θpy j px j (8) Since,x j > 0, then px j > 0. So if the inequality py j px j holds with equality, then py j = px j > 0 and θpy j < px j, which leads to a contradiction with (8). Thus, if an allocation is preferred for an individual j, then it must be more expensive than the original allocation x j. With this condition it established that x is a walrasian equilibrium with vector price p concluding the proof.